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volume 6, issue 4, article 112, 2005.

Received 14 November, 2004;

accepted 25 August, 2005.

Communicated by:L. Pick

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Journal of Inequalities in Pure and Applied Mathematics

A RELATION TO HARDY-HILBERT’S INTEGRAL INEQUALITY AND MULHOLLAND’S INEQUALITY

BICHENG YANG

Department of Mathematics Guangdong Institute of Education Guangzhou, Guangdong 510303 P. R. China.

EMail:bcyang@pub.guangzhou.gd.cn

c

2000Victoria University ISSN (electronic): 1443-5756

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A Relation to Hardy-Hilbert’s Integral Inequality and Mulholland’s Inequality

Bicheng Yang

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J. Ineq. Pure and Appl. Math. 6(4) Art. 112, 2005

http://jipam.vu.edu.au

Abstract

This paper deals with a relation between Hardy-Hilbert’s integral inequality and Mulholland’s integral inequality with a best constant factor, by using the Beta function and introducing a parameterλ.As applications, the reverse, the equivalent form and some particular results are considered.

2000 Mathematics Subject Classification:26D15.

Key words: Hardy-Hilbert’s integral inequality; Mulholland’s integral inequality; β function; Hölder’s inequality.

1. Introduction

If p > 1,¹_p + ¹_q = 1, f, g ≥ 0 satisfy 0 < R∞

0 f^p(x)dx < ∞ and 0 <

R∞

0 g^q(x)dx <∞, then one has two equivalent inequalities as (see [1]):

(1.1)

Z ∞ 0

f(x)g(y) x+y dxdy

< π sin

π p

Z ∞

0

f^p(x)dx

¹_pZ ∞ 0

g^q(x)dx ¹_q

;

(1.2)

Z ∞ 0

f(x) x+ydx

p

dy <



 π sin

π p





p

Z ∞ 0

f^p(x)dx,

where the constant factors _sin(π/p)^π and h π

sin(π/p)

ip

are all the best possible. In- equality (1.1) is called Hardy- Hilbert’s integral inequality, which is important in analysis and its applications (cf. Mitrinovic et al. [2]).

If0 <R∞ 1

1

xF^p(x)dx < ∞and0 < R∞ 1

1

yG^q(y)dy < ∞, then the Mulhol- land’s integral inequality is as follows (see [1,3]):

(1.3)

Z ∞ 1

F(x)F(y) xylnxy dxdy

< π sin

π Z ∞

1

F^p(x) x dx

¹_pZ ∞ 1

G^q(y) y dy

¹_q ,

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where the constant factor _sin(π/p)^π is the best possible. Settingf(x) =F(x)/x, andg(y) =G(y)/yin (1.3), by simplification, one has (see [12])

(1.4)

Z ∞ 1

f(x)g(y) lnxy dxdy

< π sin

π p

Z ∞

1

x^p−1f^p(x)dx

¹_pZ ∞ 1

x^q−1g^q(x)dx ¹_q

. We still call (1.4) Mulholland’s integral inequality.

In 1998, Yang [11] first introduced an independent parameter λ and theβ function for given an extension of (1.1) (forp=q= 2). Recently, by introducing a parameterλ,Yang [8] and Yang et al. [10] gave some extensions of (1.1) and (1.2) as: Ifλ >2−min{p, q}, f, g ≥0satisfy0<R∞

0 x^1−λf^p(x)dx <∞ and0<R∞

0 x^1−λg^q(x)dx <∞,then one has two equivalent inequalities as:

(1.5)

Z ∞ 0

f(x)g(y) (x+y)^λdxdy

< k_λ(p) Z ∞

0

x^1−λf^p(x)dx

¹_pZ ∞ 0

x^1−λg^q(x)dx ¹_q

and (1.6)

Z ∞ 0

y^{(p−1)(λ−1)} Z ∞

0

f(x) (x+y)^λdx

p

dy <[k_λ(p)]^p Z ∞

0

x^1−λf^p(x)dx, where the constant factors k_λ(p) and [k_λ(p)]^p (k_λ(p) = B

p+λ−2

p ,^q+λ−2_q , B(u, v)is theβfunction) are all the best possible. By introducing a parameter

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α, Kuang [5] gave an extension of (1.1), and Yang [9] gave an improvement of [5] as: If α > 0, f, g ≥ 0 satisfy 0 < R∞

0 x^{(p−1)(1−α)}f^p(x)dx < ∞ and 0<R∞

0 x^{(q−1)(1−α)}g^q(x)dx <∞,then (1.7)

Z ∞ 0

f(x)g(y) x^α+y^α dxdy

< π αsin

π p

Z ∞

0

x^{(p−1)(1−α)}f^p(x)dx

_p¹ Z ∞ 0

x^{(q−1)(1−α)}g^q(x)dx ¹_q

,

where the constant _α_sin(π/p)^π is the best possible. Recently, Sulaiman [6] gave some new forms of (1.1) and Hong [14] gave an extension of Hardy-Hilbert’s inequality by introducing two parametersλandα.Yang et al. [13] provided an extensive account of the above results.

The main objective of this paper is to build a relation to (1.1) and (1.4) with a best constant factor, by introducing the βfunction and a parameterλ,related to the double integralRb

a

Rb a

f(x)g(y)

(u(x)+u(y))^λdxdy (λ > 0).As applications, the reversion, the equivalent form and some particular results are considered.

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2. Some Lemmas

First, we need the formula of theβ function as (cf. Wang et al. [7]):

(2.1) B(u, v) :=

Z ∞ 0

1

(1 +t)^u+vt^u−1dt=B(v, u) (u, v >0).

Lemma 2.1 (cf. [4]). Ifp >1, ¹_p + ¹_q = 1, ω(σ)>0, f, g ≥0,f ∈L^p_ω(E)and g ∈L^q_ω(E),then one has the Hölder’s inequality with weight as:

(2.2) Z

E

ω(σ)f(σ)g(σ)dσ ≤ Z

E

ω(σ)f^p(σ)dσ ¹_pZ

E

ω(σ)g^q(σ)dσ ¹_q

;

if p < 1 (p 6= 0), with the above assumption, one has the reverse of (2.2), where the equality (in the above two cases) holds if and only if there exists non- negative real numbersc₁ andc₂, such that they are not all zero andc₁f^p(σ) = c₂g^q(σ),a. e. inE.

Lemma 2.2. Ifp6= 0,1, ¹_p+¹_q = 1, φ_r=φ_r(λ)>0 (r =p, q), φ_p+φ_q =λ,and u(t)is a differentiable strict increasing function in(a, b) (−∞ ≤a < b ≤ ∞) such thatu(a+) = 0andu(b−) =∞,forr=p, q,defineω_r(x)as

(2.3) ω_r(x) := (u(x))^λ−φ^r Z b

a

(u(y))^φ^r⁻¹u⁰(y)

(u(x) +u(y))^λdy (x∈(a, b)).

Then forx∈(a, b), eachω_r(x)is constant, that is (2.4) ωr(x) =B(φp, φq) (r=p, q).

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Proof. For fixedx, settingv = ^u(y)_u(x) in (2.3), one has ω_r(x) = (u(x))^λ−φ^r

Z b a

(u(y))^φ^r⁻¹u⁰(y)

(u(x))^λ(1 +u(y)/u(x))^λdy

= (u(x))^λ−φ^r Z ∞

0

(vu(x))^φ^r⁻¹

(u(x))^λ(1 +v)^λu(x)dv

= Z ∞

0

v^φ^r⁻¹

(1 +v)^λdv (r =p, q).

By (2.1), one has (2.4). The lemma is proved.

Lemma 2.3. Ifp > 1, ¹_p + ¹_q = 1, φ_r > 0 (r =p, q),satisfyφ_p+φ_q =λ,and u(t)is a differentiable strict increasing function in(a, b) (−∞ ≤a < b ≤ ∞) satisfyingu(a+) = 0andu(b−) =∞,then forc=u⁻¹(1)and0< ε < qφp,

I :=

Z b c

(u(x))^φ^q⁻^ε^p⁻¹u⁰(x)

(u(x) +u(y))^λ (u(y))^φ^p⁻^ε^q⁻¹u⁰(y)dxdy

> 1 εB

φ_p− ε

q, φ_q+ε q

−O(1) ; (2.5)

if 0 < p < 1 (or p < 0), with the above assumption and0 < ε < −qφ_q (or 0< ε < qφp), then

(2.6) I < 1

εB

φ_p −ε

q, φ_q+ε q

.

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Proof. For fixedx, settingv = ^u(y)_u(x) inI,one has I :=

Z b c

(u(x))^φ^q⁻^p^ε⁻¹u⁰(x)

"

Z b c

(u(y))^φ^p⁻^ε^q⁻¹

(u(x) +u(y))^λu⁰(y)dy

# dx

= Z b

c

(u(x))^−1−εu⁰(x) Z ∞

1 u(x)

1

(1 +v)^λv^φ^p⁻^ε^q⁻¹dvdx

= Z b

c

u⁰(x)dx (u(x))^1+ε

Z ∞ 0

v^φ^p⁻^ε^q⁻¹ (1 +v)^λdv

− Z b

c

u⁰(x) (u(x))^1+ε

Z _u(x)¹

0

v^φ^p⁻^ε^q⁻¹ (1 +v)^λdvdx (2.7)

> 1 ε

Z ∞ 0

v^φ^p⁻^ε^q⁻¹ (1 +v)^λdv−

Z b c

u⁰(x) (u(x))

"

Z _u(x)¹

0

v^φ^p⁻^ε^q⁻¹dv

# dx

= 1 ε

Z ∞ 0

v^φ^p⁻^ε^q⁻¹ (1 +v)^λdv−

φ_p− ε

q −2

.

By (2.1), inequality (2.5) is valid. If0< p <1(orp <0), by (2.7), one has I <

Z b c

u⁰(x) (u(x))^1+εdx

Z ∞ 0

1

(1 +v)^λv^φ^p⁻^ε^q⁻¹dv, and then by (2.1), inequality (2.6) follows. The lemma is proved.

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3. Main Results

Theorem 3.1. If p >1, ¹_p + ¹_q = 1, φ_r > 0 (r =p, q), φ_p+φ_q =λ, u(t)is a differentiable strict increasing function in(a, b) (−∞ ≤a < b≤ ∞),such that u(a+) = 0andu(b−) = ∞,andf, g ≥0satisfy0<Rb

a

(u(x))^{p(1−φq)−1}

(u⁰(x))^p−1 f^p(x)dx <

∞ and 0<Rb a

(u(x))^{q(1−φp)−1}

(u⁰(x))^q−1 g^q(x)dx <∞,then (3.1)

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

< B(φ_p, φ_q) Z b

a

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f^p(x)dx

1 p

× Z b

a

(u(x))^q(1−φ^p⁾⁻¹

(u⁰(x))^q−1 g^q(x)dx

1 q

, where the constant factor B(φ_p, φ_q) is the best possible. If p < 1 (p 6= 0), {λ;φr >0 (r = p, q), φp+φq = λ} 6= φ,with the above assumption, one has the reverse of (3.1), and the constant is still the best possible.

Proof. By (2.2), one has

J :=

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

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= Z b

a

Z b a

1 (u(x) +u(y))^λ

(u(x))^(1−φ^q^)/q (u(y))^(1−φ^p^)/p

(u⁰(y))^1/p (u⁰(x))^1/qf(x)

×

(u(y))^(1−φ^p^)/p (u(x))^(1−φ^q^)/q

(u⁰(x))^1/q (u⁰(y))^1/pg(y)

dxdy

≤ Z b

a

Z b a

(u(y))^φ^p⁻¹u⁰(y) (u(x) +u(y))^λ dy

(u(x))^{(p−1)(1−φ}^q⁾

(u⁰(x))^p−1 f^p(x)dx p¹

× Z b

a

Z b a

(u(x))^φ^q⁻¹u⁰(x) (u(x) +u(y))^λ dx

(u(y))^{(q−1)(1−φ}^p⁾

(u⁰(y))^q−1 g^q(y)dy ¹q

. (3.2)

If (3.2) takes the form of equality, then by (2.2), there exist non-negative num- bersc₁ andc₂, such that they are not all zero and

c₁u⁰(y)(u(x))^{(p−1)(1−φ}^q⁾

(u(y))^1−φ^p(u⁰(x))^p−1 f^p(x) = c₂u⁰(x)(u(y))^{(q−1)(1−φ}^p⁾ (u(x))^1−φ^q(u⁰(y))^q−1 g^q(y), a.e. in (a, b)×(a, b).

It follows that c₁(u(x))^p(1−φ^q⁾

(u⁰(x))^p f^p(x) = c₂(u(y))^q(1−φ^p⁾

(u⁰(y))^q g^q(y) =c₃, a.e. in (a, b)×(a, b), wherec₃ is a constant. Without loss of generality, supposec₁ 6= 0.One has

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f^p(x) = c₃u⁰(x)

c₁u(x), a.e. in (a, b),

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which contradicts0<Rb a

(u(x))^p(1−^φq)−1

(u⁰(x))^p−1 f^p(x)dx <∞.Then by (2.3), one has (3.3) J <

Z b a

ω_p(x)(u(x))^p(1−φ^q⁾⁻¹

1 p

× Z ∞

0

ωq(x)(u(x))^q(1−φ^p⁾⁻¹

(u⁰(x))^q−1 g^q(x)dx ¹_q

, and in view of (2.4), it follows that (3.1) is valid.

For0< ε < qφ_p,settingf_ε(x) =g_ε(x) = 0, x ∈(a, c) (c=u⁻¹(1));

f_ε(x) = (u(x))^φ^q⁻^ε^p⁻¹u⁰(x), g_ε(x) = (u(x))^φ^p⁻^ε^q⁻¹u⁰(x), x∈[c, b),we find

(3.4)

Z b a

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f_ε^p(x)dx ¹_p

× Z b

a

(u(x))^q(1−φ^p⁾⁻¹

(u⁰(x))^q−1 g_ε^q(x)dx ¹q

= 1 ε. If the constant factorB(φ_p, φ_q)in (3.1) is not the best possible, then, there exists a positive constant k < B(φ_p, φ_q),such that (3.1) is still valid if one replaces

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B(φ_p, φ_q)byk. In particular, by (2.6) and (3.4), one has B

φ_p− ε

q, φ_q+ ε q

−εO(1)

< ε Z b

a

Z b a

f_ε(x)g_ε(y)

(u(x) +u(y))^λdxdy

< εk Z b

a

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f_ε^p(x)dx

¹_p Z b a

(u(x))^q(1−φ^p⁾⁻¹

(u⁰(x))^q−1 g^q_ε(x)dx ¹_q

=k, and thenB(φ_p, φ_q)≤k(ε→0⁺).This contradicts the fact thatk < B(φ_p, φ_q).

Hence the constant factorB(φ_p, φ_q)in (3.1) is the best possible.

For 0 < p < 1 (or p < 0), by the reverse of (2.2) and using the same procedures, one can obtain the reverse of (3.1). For0< ε <−qφ_q(or0< ε <

qφ_p), setting f_ε(x) andg_ε(x)as the above, we still have (3.4). If the constant factorB(φ_p, φ_q)in the reverse of (3.1) is not the best possible, then, there exists a positive constantK > B(φ_p, φ_q),such that the reverse of (3.1) is still valid if one replacesB(φ_p, φ_q)byK. In particular, by (2.7) and (3.4), one has

B

φ_p− ε

q, φ_q+ ε q

> ε Z b

a

Z b a

f_ε(x)g_ε(y)

(u(x) +u(y))^λdxdy

> εK Z b

a

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f_ε^p(x)dx

¹pZ b a

(u(x))^q(1−φ^p⁾⁻¹

(u⁰(x))^q−1 g_ε^q(x)dx ¹q

=K,

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and then B(φ_p, φ_q) ≥ K (ε → 0⁺). This contradiction concludes that the constant in the reverse of (3.1) is the best possible. The theorem is proved.

Theorem 3.2. Let the assumptions of Theorem3.1hold.

(i) If p > 1,¹_p + ¹_q = 1, one obtains the equivalent inequality of (3.1) as follows

(3.5) Z b

a

u⁰(y) (u(y))^1−pφ^p

Z b a

f(x)

(u(x) +u(y))^λdx ^p

dy

<[B(φ_p, φ_q)]^p Z b

a

(u(x))^p(1−φ^q⁾⁻¹

(u⁰(x))^p−1 f^p(x)dx;

(ii) If 0< p < 1,one obtains the reverse of (3.5) equivalent to the reverse of (3.1);

(iii) Ifp < 0,one obtains inequality (3.5) equivalent to the reverse of (3.1), where the constants in the above inequalities are all the best possible.

Proof. Set

g(y) := u⁰(y) (u(y))^1−pφ^p

Z b a

f(x)

(u(x) +u(y))^λdx ^p−1

,

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and use (3.1) to obtain 0<

Z b a

(u(y))^q(1−φ^p⁾⁻¹

(u⁰(y))^q−1 g^q(y)dy

= Z b

a

u⁰(y) (u(y))^1−pφ^p

Z b a

f(x)

dy

= Z b

a

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy ≤B(φp, φq)

× Z b

a

(u(x))^p(1−φ^q⁾⁻¹

1 p

× Z b

a

(u(y))^q(1−φ^p⁾⁻¹

1 q

; (3.6)

0<

Z b a

(u(y))^q(1−φ^p⁾⁻¹

(u⁰(y))^q−1 g^q(y)dy ¹⁻¹q

= (Z b

a

u⁰(y) (u(y))^1−pφ^p

Z b a

f(x)

dy )¹_p

≤B(φ_p, φ_q) Z b

a

(u(x))^p(1−φ^q⁾⁻¹

1 p

<∞.

(3.7)

It follows that (3.6) takes the form of strict inequality by using (3.1); so does

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(3.7). Hence one can get (3.5). On the other hand, if (3.5) is valid, by (2.2), Z b

a

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

= Z b

a

"

(u⁰(y))¹^p (u(y))¹^p^−φ^p

Z b a

f(x)

(u(x) +u(y))^λdx

# "

(u(y))¹^p^−φ^p (u⁰(y))¹^p

g(y)

# dy

≤ (Z b

a

u⁰(y) (u(y))^1−pφ^p

Z b a

f(x)

dy )¹_p

× Z b

a

(u(y))^q(1−φ^p⁾⁻¹

1 q

(3.8) .

Hence by (3.5), (3.1) yields. It follows that (3.1) and (3.5) are equivalent.

If the constant factor in (3.5) is not the best possible, one can get a contradiction that the constant factor in (3.1) is not the best possible by using (3.8).

Hence the constant factor in (3.5) is still the best possible.

If0 < p < 1(orp < 0), one can get the reverses of (3.6), (3.7) and (3.8), and thus concludes the equivalence. By (3.6), for0< p <1,one can obtain the reverse of (3.5); forp <0,one can get (3.5). If the constant factor in the reverse of (3.5) (or simply (3.5)) is not the best possible, then one can get a contradiction that the constant factor in the reverse of (3.1) is not the best possible by using the reverse of (3.8). Thus the theorem is proved.

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4. Some Particular Results

We point out that the constant factors in the following particular results of The- orems3.1–3.2are all the best possible.

4.1. The first reversible form

Corollary 4.1. Let the assumptions of Theorems3.1–3.2hold. For

φr =

1− 1 r

(λ−2) + 1 (r=p, q),

0<

Z b a

(u(x))^1−λ

(u⁰(x))^p−1f^p(x)dx <∞ and

0<

Z b a

(u(x))^1−λ

(u⁰(x))^q−1g^q(x)dx <∞, settingkλ(p) =B

p+λ−2

p ,^q+λ−2_q

,

(i) If p >1, ¹_p + ¹_q = 1, λ > 2−min{p, q}, then we have the following two equivalent inequalities:

(4.1) Z b

a

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

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< k_λ(p) Z b

a

(u(x))^1−λ

(u⁰(x))^p−1f^p(x)dx

¹_pZ b a

(u(x))^1−λ

(u⁰(x))^q−1g^q(x)dx ¹_q

and

(4.2) Z b

a

u⁰(y) (u(y))^{(p−1)(1−λ)}

Z b a

f(x)

dy

<[k_λ(p)]^p Z b

a

(u(x))^1−λ

(u⁰(x))^p−1f^p(x)dx.

(ii) If0< p <1and2−p < λ <2−q, one obtains two equivalent reverses of (4.1) and (4.2),

(iii) If p < 0 and 2−q < λ < 2−p, we have the reverse of (4.1) and the inequality (4.2), which are equivalent. In particular, by (4.1),

(a) settingu(x) =x^α (α >0, x∈(0,∞)),one has (4.3)

Z ∞ 0

f(x)g(y) (x^α+y^α)^λdxdy

< 1 αk_λ(p)

Z ∞ 0

xp−1+α(2−λ−p)

f^p(x)dx ¹_p

× Z ∞

0

xq−1+α(2−λ−q)

g^q(x)dx ¹_q

;

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(b) settingu(x) = lnx, x∈(1,∞),one has (4.4)

Z ∞ 1

f(x)g(y) (lnxy)^λ dxdy

< kλ(p) Z ∞

1

x^p−1(lnx)^1−λf^p(x)dx ¹_p

× Z ∞

1

x^q−1(lnx)^1−λg^q(x)dx ¹_q

;

(c) settingu(x) =e^x, x ∈(−∞,∞),one has (4.5)

Z ∞

−∞

Z ∞

−∞

f(x)g(y) (e^x+e^y)^λdxdy

< k_λ(p) Z ∞

−∞

e^{(2−p−λ)x}f^p(x)dx ¹_p

× Z ∞

−∞

e^{(2−q−λ)x}g^q(x)dx ¹_q

;

(d) settingu(x) = tanx, x∈(0,^π₂),one has (4.6)

Z ^π₂

0

Z ^π₂

0

f(x)g(y)

(tanx+ tany)^λdxdy

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< k_λ(p) (Z ^π₂

0

tan^1−λx

sec^2(p−1)xf^p(x)dx )¹_p(

Z ^π₂

0

tan^1−λx

sec^2(q−1)xg^q(x)dx )¹_q

;

(e) settingu(x) = secx−1, x∈(0,^π₂),one has (4.7)

Z ^π₂

0

Z ^π₂

0

f(x)g(y)

(secx+ secy−2)^λdxdy

< k_λ(p) (Z ^π₂

0

(secx−1)^1−λ

(secxtanx)^p−1f^p(x)dx )_p¹

× (Z ^π₂

0

(secx−1)^1−λ

(secxtanx)^q−1g^q(x)dx )¹_q

.

4.2. The second reversible form

Corollary 4.2. Let the assumptions of Theorems3.1–3.2hold. For

φ_r = λ−1 2 +1

r (r=p, q), 0<

Z b a

(u(x))^p^1−λ²

(u⁰(x))^p−1f^p(x)dx <∞ and

0<

Z b (u(x))^q^1−λ²

g^q(x)dx <∞,

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settingkeλ(p) =B

pλ−p+2

2p ,^qλ−q+2_2q

,

(i) Ifp >1,¹_p+¹_q = 1, λ >1−2 min{¹_p,¹_q}, then one can get two equivalent inequalities as follows:

(4.8) Z b

a

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

<ke_λ(p) (Z b

a

(u(x))^p^1−λ²

(u⁰(x))^p−1f^p(x)dx )¹_p

× (Z b

a

(u(x))^q^1−λ²

(u⁰(x))^q−1g^q(x)dx )¹_q

;

(4.9) Z b

a

u⁰(y) (u(y))^p

1−λ 2

Z b a

f(x)

dy

<h

ke_λ(p)ipZ b a

(u(x))^p^1−λ²

(u⁰(x))^p−1f^p(x)dx, (ii) If0< p <1,1−_p² < λ <1−²_q,one can get two equivalent reversions of

(4.8) and (4.9),

(iii) If p < 0, 1− ²_q < λ < 1− ²_p, one can get the reversion of (4.8) and inequality (4.9), which are equivalent. In particular, by (4.8),

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(a) settingu(x) =x^α (α >0, x∈(0,∞)),one has (4.10)

Z ∞ 0

< 1 αkeλ(p)

Z ∞ 0

x^{p−1+α(1−p}^1+λ² ⁾f^p(x)dx ¹_p

× Z ∞

0

x^{q−1+α(1−q}^1+λ² ⁾g^q(x)dx ¹_q

;

Z ∞ 1

<ke_λ(p) Z ∞

1

x^p−1(lnx)^p^1−λ² f^p(x)dx ¹_p

× Z ∞

1

x^q−1(lnx)^q^1−λ² g^q(x)dx ¹_q

;

(c) settingu(x) =e^x, x ∈(−∞,∞),one has (4.12)

Z ∞

−∞

Z ∞

−∞

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<ke_λ(p) Z ∞

−∞

e^(1−p^1+λ² ^)xf^p(x)dx

¹_pZ ∞

−∞

e^(1−q^1+λ² ^)xg^q(x)dx ¹_q

;

(d) settingu(x) = tanx, x∈(0,^π₂),one has

(4.13) Z ^π₂

0

Z ^π₂

0

f(x)g(y)

(tanx+ tany)^λdxdy

<ke_λ(p) (Z ^π₂

0

tan^p^1−λ² x

sec^2(p−1)xf^p(x)dx )¹_p

× (Z ^π₂

0

tan^q^1−λ² x

sec^2(q−1)xg^q(x)dx )¹_q

;

(e) settingu(x) = secx−1, x ∈(0,^π₂),one has (4.14)

Z ^π₂

0

Z ^π₂

0

f(x)g(y)

<ke_λ(p) (Z ^π₂

0

(secx−1)^p^1−λ²

(secxtanx)^p−1f^p(x)dx )_p¹

× (Z ^π₂

0

(secx−1)^q^1−λ²

(secxtanx)^q−1g^q(x)dx )¹_q

.

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4.3. The form which does not have a reverse

Corollary 4.3. Let the assumptions of Theorems3.1–3.2hold. For φ_r= λ

r(r=p, q), if p > 1,1 p +1

q = 1, λ >0, 0<

Z b a

(u(x))^{(p−1)(1−λ)}

(u⁰(x))^p−1 f^p(x)dx <∞ and

0<

Z b a

(u(x))^{(q−1)(1−λ)}

(u⁰(x))^q−1 g^q(x)dx <∞, then one can get two equivalent inequalities as:

(4.15) Z b

a

Z b a

f(x)g(y)

(u(x) +u(y))^λdxdy

< B λ

p,λ q

Z b a

(u(x))^{(p−1)(1−λ)}

(u⁰(x))^p−1 f^p(x)dx ¹_p

× Z b

a

(u(x))^{(q−1)(1−λ)}

(u⁰(x))^q−1 g^q(x)dx ¹_q

;

(4.16) Z b

a

u⁰(y) (u(y))^1−λ

Z b a

f(x)

dy

<

B

λ p,λ

q

pZ b (u(x))^{(p−1)(1−λ)}

(u⁰(x))^p−1 f^p(x)dx.

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In particular, by (4.15),

(a) settingu(x) =x^α(α >0;x∈(0,∞)),one has (4.17)

Z ∞ 0

< 1 αB

λ p,λ

q

Z ∞ 0

x(p−1)(1−αλ)

f^p(x)dx ¹_p

× Z ∞

0

x(q−1)(1−αλ)

g^q(x)dx ¹_q

;

Z ∞ 1

< B λ

p,λ q

Z ∞ 1

x^p−1(lnx)^{(p−1)(1−λ)}f^p(x)dx ¹_p

× Z ∞

1

x^q−1(lnx)^{(q−1)(1−λ)}g^q(x)dx ¹_q

;

(c) settingu(x) =e^x, x∈(−∞,∞),one has (4.19)

Z ∞

−∞

Z ∞

−∞

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< B λ

p,λ q

Z ∞

−∞

e^(1−p)λxf^p(x)dx

¹_pZ ∞

−∞

e^(1−q)λxg^q(x)dx ¹_q

;

(d) settingu(x) = tanx, x∈(0,^π₂),one has

(4.20) Z ^π₂

0

Z ^π₂

0

f(x)g(y)

(tanx+ tany)^λdxdy

< B λ

p,λ q

( Z ^π₂

0

tan^{(p−1)(1−λ)}x

sec^2(p−1)x f^p(x)dx )¹_p

× (Z ^π₂

0

tan^{(q−1)(1−λ)}x

sec^2(q−1)x g^q(x)dx )¹_q

;

(e) settingu(x) = secx−1, x∈(0,^π₂),one has (4.21)

Z ^π₂

0

Z ^π₂

0

f(x)g(y)

< B λ

p,λ q

( Z ^π₂

0

(secx−1)^{(p−1)(1−λ)}

(secxtanx)^p−1 f^p(x)dx )¹_p

× (Z ^π₂

0

(secx−1)^{(q−1)(1−λ)}

(secxtanx)^q−1 g^q(x)dx )¹_q

.

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Remark 1. For α = 1, (4.3) reduces to (1.5). For λ = 1, inequalities (4.3), (4.10) and (4.17) reduce to (1.7), and inequalities (4.4), (4.11) and (4.18) reduce to (1.4). It follows that inequality (3.5) is a relation between (1.4) and (1.7)(ro (1.1)) with a parameterλ.Still forλ = 1,(4.5), (4.12) and (4.19) reduce to (4.22)

Z ∞

−∞

Z ∞

−∞

f(x)g(y) e^x+e^y dxdy

< π sin

π p

Z ∞

−∞

e^(1−p)xf^p(x)dx

_p¹ Z ∞

−∞

e^(1−q)xg^q(x)dx ¹_q

, (4.6), (4.13) and (4.20) reduce to

(4.23) Z ^π₂

0

Z ^π₂

0

f(x)g(y)

tanx+ tanydxdy

< π sin

π p

(Z ^π₂

0

cos^2(p−1)xf^p(x)dx )¹_p(

Z ^π₂

0

cos^2(q−1)xg^q(x)dx )¹_q

, and (4.7), (4.14) and (4.21) reduce to

(4.24) Z ^π₂

0

Z ^π₂

0

f(x)g(y)

secx+ secy−2dxdy

< π sin

π p

(Z ^π₂

0

cos²x sinx

^p−1

f^p(x)dx )¹_p(

Z ^π₂

0

cos²x sinx

^q−1

g^q(x)dx )¹_q

.

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References

[1] G. H. HARDY, J.E. LITTLEWOOD ANDG. POLYA, Inequalities, Cam- bridge University Press, Cambridge, 1952.

[2] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Boston, 1991.

[3] H.P. MULHOLLAND, Some theorems on Dirichlet series with positive coefficients and related integrals, Proc. London Math. Soc., 29(2) (1929), 281–292.

[4] JICHANG KUANG, Applied Inequalities, Shangdong Science and Tech- nology Press, Jinan, 2004.

[5] JICHANG KUANG, On a new extension of Hilbert’s integral inequality, J. Math. Anal. Appl., 235 (1999), 608–614.

[6] W.T. SULAIMAN, On Hardy-Hilbert’s integral inequality, J. Inequal. in Pure and Appl. Math., 5(2) (2004), Art. 25. [ONLINE:http://jipam.

vu.edu.au/article.php?sid=385]

[7] ZHUXI WANG AND DUNRIN GUO, An Introduction to Special Func- tions, Science Press, Beijing, 1979.

[8] BICHENG YANG, On a general Hardy-Hilbert’s inequality with a best value, Chinese Annals of Math., 21A(4) (2000), 401–408.

[9] BICHENG YANG, On an extension of Hardy-Hilbert’s inequality, Chi-